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In Chapter 14 , we have observed some limitations in the use of the Fourier transform. For instance, in Section 14.3 , we have obtained that the Fourier transform of \(H(t)\,\mathrm {e}^{\beta t}\) exists only for \(\beta <0\) (Eq. 14.44 ). Similarly, in Section 14.8 we have seen that the Fourier transform of the impulse response for a second–order differential equation, namely \(h(t)=H(t)\,\mathrm {e}^{\beta {t}}\sin \omega {t}/\omega \) (Eq. 14.144 ), exists only for \(\beta <0\) . These drawbacks are quite limiting for practical applications.

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Laplace transform

  • 章节

  • 发表日期：

    2026年

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  • Luigi Morino

摘要

In Chapter 14 , we have observed some limitations in the use of the Fourier transform. For instance, in Section 14.3 , we have obtained that the Fourier transform of \(H(t)\,\mathrm {e}^{\beta t}\) exists only for \(\beta <0\) (Eq. 14.44 ). Similarly, in Section 14.8 we have seen that the Fourier transform of the impulse response for a second–order differential equation, namely \(h(t)=H(t)\,\mathrm {e}^{\beta {t}}\sin \omega {t}/\omega \) (Eq. 14.144 ), exists only for \(\beta <0\) . These drawbacks are quite limiting for practical applications.